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10

Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends) $$ \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 U}{\partial S^2}(S,t)+\frac{\partial U}{\partial t}(S,t)=0 $$ Let ${\cal V}(S,t) = \frac{\partial U}{\partial \sigma}(S,t)$ be the option vega. Differentiating ...


9

Hints: You know the vega of a digital call option formula: $V=-\frac{e^{-r(T-t)}}{\sigma} d_1 n\left(d_2\right)$ Where n is the standard normal density, which is positive. Sigma and exponential are also positive, so the sign of V is down to the sign of $d_1$. Which is negative when: $d_1 <0$ $\ln \frac{S}{X}+\left(r+0.5\sigma^2\right)(T-t)<0$ $S&...


8

We assume that the process $\{r(t), \, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dr(t) = \big( \theta(t) - a(t) r(t) \big)dt + \sigma(t) dW_t, \quad t > 0, \end{align*} where $\{W_t, \, t \ge 0 \}$ is a standard Brwonian motion. Note that \begin{align*} d\left(e^{\int_0^t a(u) du}r(t) \right) &=a(t) e^{\int_0^t a(u) du}r(t) dt + e^{\int_0^...


7

At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with assuming arbitrage is not possible (otherwise it would be trivial wouldn't it). But, in my opinion, what you actually seek is the Efficient Markets Hypothesis....


6

It is covered very nicely in Iain Clark's Foreign Exchange Option Pricing, A Practitioner’s Guide (pages 98-104). The book also contains references to the relevant literature including Feller's original paper.


6

Using the distribution and independence of increments allows to prove $L^2$ (mean-square) continuity. Proving the a.s. continuity is much harder. Paul Lévy's construction of Brownian motion is related in Le Gall; an alternative is to construct the Brownian motion through Haar wavelet functions or Fourier series.


6

I'll give a heuristic "proof" for general European claims which will cause mathematicians to feel sick, but which physicists / practitioners would probably be quite happy work with: Write the Black-Scholes PDE as $$ \frac{\partial F}{\partial\tau}(\tau) = \mathcal{A} F(\tau) $$ with $\tau = T- t$, and the operator $\mathcal A$ is defined as $$ \...


5

This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between ...


5

Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now, $d$ is an interest rate, and the condition suggests that $d$ is too high. So you will want to receive $d$ in order to profit. If you could, you would borrow money at $r$ and lend it to the stock broker or exchange to collect the interest ...


5

What you need to do is to first make a variable change such as $u = \frac{x-\xi}{2\sqrt{t}}$. Then change the order of the limit and the integral.


4

The paper could be clearer indeed. It is a slightly confusing topic, but the important step here is to understand the consequence of the derivative $C$ in the portfolio being priced at the assumed vol $\sigma$. This implies (by Black-Scholes) that it will by definition be true that: $\theta_t + \frac{\partial{C}}{\partial{S}}rS_t+ \frac{1}{2}\frac{\partial^...


4

The whole point of no-arbitrage pricing in a complete market is that a general underlying model of the form $$d S_t = \mu(S_t,t)\, dt + \sigma(S_t,t) \, dW_t$$ can be replaced with the risk-neutral process. $$d S_t = (r - \sigma^2/2)\, dt + \sigma(S_t,t) \, dW_t$$ for the purpose of finding the theoretical fair option price. This, of course, follows ...


4

Consider an (arithmetic) Ornstein-Uhlenbeck process as a model of the asset price $X_t$: $$dX_t = \kappa(\mu-S_t)dt + \sigma dW_t$$ where $\mu$ is the mean-reversion level, $\sigma$ is a volatility parameter, $W_t$ is Brownian motion, and $\kappa$ is the reversion speed. An Ornstein-Uhlenbeck process will revert to the mean infinitely often if $\kappa >...


4

$X_j$ can be either 1 or -1 with 50% probability each. So this step is just applying the expectation to both possible cases. See definition of the Expectation... \begin{align} {\mathbb E}\bigl[ X \bigr] = \sum_i i \cdot p(x = i) \end{align} It's the sum over all possibilities of the probability of getting that value (both ${\frac 1 2}$ in your case) ...


4

OK, here is a simplified demonstration: Before we consider swaps, let us consider very simple bonds. Suppose that you have a choice of two zero-coupon bonds. A riskless one costs 95 and is certain to pay 100 in 1 year. A risky one costs 90, is expected to also pay 100 in 1 year, but with some probability $p$ will default and only pay some $R<100$ on the ...


3

A very good book covering such fundamentals with no or only a minimal amount of maths — highly recommended! Puzzles of Finance: Six Practical Problems and Their Remarkable Solutions by Mark P. Kritzman The topics that are covered here are: Siegel's Paradox Likelihood of Loss Time Diversification Why the Expected Return Is Not To Be Expected Half Stocks ...


3

Write out the simple equations $$\begin{align} Y_j &= a_0 Z_j + a_1 Z_{j-1} + a_2 Z_{j-2}\\ Y_{j-1} &= a_0 Z_{j-1} + a_1 Z_{j-2} + a_2 Z_{j-3} \end{align}$$ There are some very simple cases that make $Y_j \perp Y_{j-1}$ due to the independence assumption of the random variables $\{Z_i\}_{i\in\mathbb{Z}}$. An example is $a_0 \in \mathbb{R}\setminus \...


3

overall gamma is second derivative of whole portfolio over underlying. adding any function (such as underlying*constant) which second derivative is 0 does not alter overall second derivative.


3

I believe your SDE has an unintended error. It should be: $$ dr_t = a \cdot (b - r_t) \cdot dt + \sigma \cdot \sqrt{r_t} \cdot dB_t. $$ On the other hand, the Feller condition is discussed and explained in Section 10.2.1.2 (pg. 432) of the Andersen and Piterbarg book: Interest Rate Modeling. Hope it helps!


3

It's part of the definition I'd just like to re-iterate the comment by Kevin, (which as far as I can tell is the answer). There are three properties which define a standard Brownian motion / Wiener process: Independent increments. Normally distributed with variance equal to the time increment. The path is continuous. Which hopefully any "standard"...


2

Why do you think such a theorem would exist? I will give you a counterexample: You have two assets A and B. Both are completely identical in every respect, except price: The price of A is USD 1 the price of B is USD 2. Your strategy is simple: You (short) sell B for a gain of 2 and buy A for 1. This strategy requires no capital and leaves you with an ...


2

Let $X$ be endowed with the following partial order: $y \geq x $ means that $\Bbb P(y\geq x) = 1$. The AOA condition in your case states that the pricing law $p$ is strictly inctreasing with respect to $\geq$, whereas LOP says that $p$ is a linear function. Neither if the two implies another one in general. For example, if $X = \Bbb R$ then $p(x) = x^3$ is a ...


2

This doesn't really suffice as an existence proof, but you can start with a series of mathematical results collectively known as no free lunch theorems. The linked paper proves the average performance of any optimization algorithm over arbitrary problem domains is independent of the algorithm. That is, no single algorithm can ever be better than others on ...


2

This is where one needs the concept of no free lunch with vanishing risk (NFLVR), whose proof you can find in: Delbaen & Schachermayer (1994). Though, as a warning, I should mention it is pretty involved.


2

The comments above re all the entries of $\mu$ not being the same is true, but can be removed if you make the 2x2 determinant in question $\ge 0$ instead of $> 0$. The commenters know this of course. The answer to your question can be obtained by an application of the Cauchy-Schwartz inequality along with knowledge that a symmetric positive definite ...


2

In a local volatility model, which is inhomogeneous in space, you'll end up with having that implied volatility of a vanilla option $(K,T)$ is a function of the spot price $S$, i.e. $$ \Sigma = \sigma(T,K,S) $$ As such when you compute the (total) derivative of the option price with respect to the spot price you'll have: \begin{align} \frac{d V}{d S} &= \...


2

If you can prove that $\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})=t\ln(\frac{E[e^{t^{-1}X}]}{\alpha})$ is convex and apply the property of positive homogeneity, then the sub-additivity follows. In original paper, authors show that $\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})$ is convex in $(X,t)$. Lemma: For fixed $\alpha$, all $\lambda\in[0,1],X,Y\in L_{M^+}$ ...


2

$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\V}{\mathbb{V}}$ First note that \begin{eqnarray} \E[(r_{t + n} + 1)^2] &=& \E[r_{t+n}^2 + 2r_{t + n} + 1] = \E[r_{t + n}^2] + 2\mu + 1 \\&=& (\V[r_{t+n}] + \E^2[r_{t+n}]) + 2 \mu + 1 \\ &=& \sigma^2 + (\mu^2 + 2\mu + 1) = \sigma^2 + (\mu + 1)^2 \tag{1} \end{eqnarray} Now, since $r_{t+n}$...


2

Let $p\in(0,1)$. The corresponding quantile function of $X\sim N(\mu,\sigma^2)$ is given by $$F_X^{-1}(p)=\mu+\sigma\Phi^{-1}(p)=\mu+\sqrt{2}\sigma\mathrm{erf}^{-1}(2p-1),$$ where $\Phi^{-1}$ is the inverse of the cumulative distribution function of a standard normally distributed random variable and $\mathrm{erf}^{-1}$ is the inverted error function. Thus, ...


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